The relationship between instantaneous frequency and time-frequency representations

We give the relationship between instantaneous frequency estimation via the derivative of the phase of the analytic signal and the first moment of general time-frequency representations from Cohen’s Manuscript received March, 2, 1990; revised January 25, 1992. This work was supported by the Australian Research Council. B. C. Lovell and B. Boashash are with the Department of Electrical Engineering, University of Queensland, St. Lucia, Brisbane, Australia 4067. R. C. Williamson is with the Department of Systems Engineering and Interdisciplinary Program in Engineering, Australian National University, Canberra, Australia 2601. IEEE Log Number 9206002. . , 1053-587X/93$03.00 @ 1993 IEEE class in both the continuous and discrete-time domains. Many researchers have applied the standard linear definition of first moment to discrete-time time-frequency representations although this leads to biased instantaneous frequency estimators with high variance; we show that periodic (circular) definitions of moments must be used to account for the periodization of the frequency variable due to sampling.